System and method for analyzing risk

ABSTRACT

The present invention provides a method and system for analyzing risk. An embodiment of this invention comprises determining a deployment cost and an outage cost for each of a plurality of portfolios. The portfolio comprises one or more elements with a predetermined capacity. Total cost, which is an indicator of deployment cost and risk, for each portfolio is then determined by summing the deployment cost and the outage cost. The total cost of at least two portfolios is compared.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. §119(e) to U.S.Provisional Application No. 60/386,832, titled “Probabilistic Model ToAnalyze Risks Associated With Telecommunication Network Size,” filedJun. 6, 2002, the entirety of which is hereby incorporated by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention generally relates to risk analysis. More specifically,this invention relates to a system and method for analyzing the risksassociated with implementing telecommunication networks of varyingnetwork element sizes.

2. Related Art

A common issue facing large telecommunications carriers involvesdetermining the optimum size for telecommunications network elements.Due to economies of scale and the development of new technologies, manycarriers utilize infrastructure components of ever increasing size. Thisresults in concentrating larger amounts of customer traffic onto fewerresources. For examples, telecommunication carriers utilize largercircuit switches and cross-connects and fibers with greater capacity.However, the concentration of larger amounts of traffic on fewerresources means that the failure of such a resource can have alarge—even catastrophic—impact.

Thus, a problem facing telecommunications carriers is determining thesize at which the risks associated with larger network elements overtakethe cost savings provided by larger network elements. A need exists fora model that analyzes this issue.

SUMMARY OF THE INVENTION

An embodiment of the present invention provides a model for analyzingnetwork elements, preferably of a single type (e.g., a digitalcross-connect). A purpose of this model is to compare scenarios thatdiffer only in the size of the network elements, in order to provide ananalytical framework for determining an optimum size for the networkelements that make up a network.

In one embodiment, the present invention provides a method for analyzingrisk. This method comprises determining a deployment cost and an outagecost for each of a plurality of portfolios. The portfolio comprises oneor more elements with a predetermined capacity. Total cost, which is anindicator of deployment cost and risk, for each portfolio is thendetermined by summing the deployment cost and the outage cost. The totalcost of at least two portfolios is compared. Deployment costs may bedetermined responsive to the cost of obtaining each element andrecurring costs associated with maintaining each element. Outage costsare determined responsive to the probability of an element outage andthe direct and indirect costs of that outage.

In additional embodiments, each element within a particular portfoliohas identical capacity.

In another embodiment, the present invention again provides a method foranalyzing risk. Once again, this method comprises determining adeployment cost and an outage cost for each of a plurality ofportfolios. The portfolio comprises one or more elements with apredetermined capacity. Total cost for each portfolio is then determinedby summing the deployment cost and the outage cost. Moreover, a totalcost variability is determined for each portfolio. Expected utility isdetermined responsive to total cost and total cost variability. In someembodiments, the portfolio with the highest expected utility may beselected as the optimum portfolio.

In another embodiment, a computer program product is provided comprisinga computer usable medium having computer program logic recorded thereonfor instructing a computer system to determine deployment and outagecosts, total cost and total cost variability, expected utility and tocompare expected utility between at least two portfolios.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts a method according to an embodiment of the presentinvention for analyzing risk.

FIG. 2 depicts an example demand profile for a central office (CO).

FIG. 3 depicts an example representation of the number of digital crossconnects (DCS) for a number of portfolios.

FIG. 4 depicts an example deployment cost graph.

FIG. 5 depicts an example comparison of ripple effect costs.

FIG. 6 depicts an example average outage cost graph.

FIG. 7 depicts an example average lifecycle total cost graph.

FIG. 8 depicts an example graph of outage cost standard deviation.

FIG. 9 depicts a series of example histograms of lifecycle costs.

FIG. 10 depicts an example utility function.

FIG. 11 depicts example expected utilities.

FIG. 12 depicts a method according to an embodiment of the presentinvention for analyzing risk.

FIG. 13 depicts a method according to an embodiment of the presentinvention for analyzing risk.

FIG. 14 depicts a block diagram of an example computer system.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Reference now will be made to the present preferred embodiments of theinvention, examples of which are illustrated in the accompanyingdrawings.

In one embodiment, the present invention provides a model for analyzingnetwork elements, preferably of a single type (e.g., a digitalcross-connect). A purpose of this model is to compare scenarios thatdiffer only in the size of the network elements, in order to provide ananalytical framework for determining an optimum size for the networkelements that make up a network. Although the present invention has beendescribed herein with reference to analyzing network elements, it shouldbe understood that this is by way, of example only, and the presentinvention may be applied in other embodiments not specifically describedherein.

A preferred embodiment of the present invention is shown in FIG. 1. Inthe first step of the method shown in FIG. 1, step 101, the deploymentcost for several portfolios is determined. As used herein, a portfoliois a collection of elements that serve some purpose. For example, aportfolio can be a collection of network elements that serve totalcustomer demands that are the same for all portfolios. Portfolios differin the size and number of the network elements they contain. Sincedemand is independent of portfolio, a portfolio of larger elements hasfewer elements. Both the demand on all portfolios and the number ofelements in each portfolio can change over time. In a preferredembodiment, each network element within a portfolio is of equal size orcapacity.

The deployment cost, which is determined in step 101, may include bothcapital cost associated with the installation of equipment and theongoing operating costs. In general, given a demand forecast, thedeployment costs are fairly straightforward and predictable. In apreferred embodiment, deployment cost is the present value of all suchcosts that occur in a specified time interval. In general, deploymentcosts benefit from economies of scale. Thus, a portfolio's deploymentcost per circuit (and also its overall deployment cost because demand incircuits is the same for all portfolios) may decrease as the networkelement size increases.

In the next step, step 102, the outage cost of the portfolio isdetermined. Since outages are random in both occurrence and impact, theoutage cost is preferably a random variable, which is expressed in aunit of measure comparable to deployment cost. For example, outage (anddeployment) cost may be expressed in dollars. The portfolio outage costdistribution may have a longer tail as the network element size growsbecause larger outages generally have proportionally larger costs.

In the third step in the method shown in FIG. 1, step 103, a portfolio'stotal cost is determined. Total cost is determined by summing deploymentcost and outage cost over a specified time interval, such as a lifecycle.

Because in general a portfolio's deployment cost decreases with elementsize but its outage cost increases, the behavior of the average totalcost with respect to element size is unpredictable. Thus, in step 104,the total cost of at least two portfolios is compared in order todetermine which of the two portfolios has the lower total cost.

It is possible that there is a threshold beyond which the element is“too big.” In other words, beyond a certain size, some statistic of thetotal portfolio cost becomes unacceptably large. For example, the meanor the probability of a catastrophic outage might become too large.

In a preferred embodiment, the present invention will provide a modelfor reconciling the savings due to economies of scale against the risksin the tails of the outage cost distribution by using stochasticdominance and utility functions.

The following discusses the determination of deployment cost in moredetail. In particular, the deployment cost associated with a network ofbroadband digital cross-connects (DCS) is discussed. However, this isfor illustration purposes only and the present invention is not limitedto this context.

A broadband DCS is a sorting machine that resides in a central office(CO). It unpacks and repacks high-speed transport facilities (OC3, OC12,OC48, etc.) in units of DS3's. That is, the DCS can demultiplex DS3'sout of its high-speed incoming interfaces and recombine the DS3's tofeed outgoing high-speed interfaces. Circuits terminate on ports; thecross-connect's switching matrix accomplishes the sorting and bundlingof DS3's onto outgoing interfaces. Vendors usually express the size ofthe matrix in terms of DS3-equivalent ports. Since each circuit uses twoports per machine, a DCS with a 6000 DS3-equivalent port matrix cancarry 3000 DS3-equivalent circuits.

Broadly speaking, a DCS consists of “common” equipment, that allcircuits share, and ports, that serve one circuit each. Examples ofcommon equipment are the control and administrative complex in a DCS andthe matrix. The presence of fixed costs means that if the capacity ofthe common equipment is doubled, its cost increases by less than afactor of two. Symbolically, if e denotes the common equipment cost as afunction of its capacity c in circuits and r is a positive integer, thene(rc)≦re(c).

The last inequality suggests that r machines of circuit capacity c andone machine of circuit capacity rc can serve the same demands. In fact,the circuit capacity of one large machine may be larger than that of ther smaller machines because connections between machines use up capacity.When no port to a destination is available to an incoming circuit on theDCS, but such a port is available on another DCS in the CO, the circuitmust go from the first DCS to the second and then to its destination.The connections between DCS's, called tie pairs, typically use about 10%of the ports on each DCS. A circuit that traverses only one DCS uses twoports in the CO; a circuit that traverses a tie pair uses a total offour ports. Because of this, r DCS's, each of circuit capacity c, canhandle fewer than rc circuits. A portfolio of fewer, larger DCS'sreduces tie pair waste, thereby saving cost.

Bigger network elements also reduce ongoing expenses. Network elementsrequire software upgrades at least annually. A portfolio with fewermachines requires fewer software upgrades. Assuming that the price permachine for software is independent of machine size, the total annualsoftware cost is proportional to the number of machines in theportfolio.

Moreover, as the circuit demand grows, the carrier must add DCS's morefrequently the smaller the machines are. Each addition entails planningactivities, as well as testing and turn-up. However, the associatedexpenses are small compared to the costs of the activities discussed sofar and, in a preferred embodiment, may be neglected.

In addition, it may be expected that provisioning, monitoring andmaintenance is more costly with a larger number of smaller machines,especially with tie pairs deployed. However, the advantages of thelarger-machine portfolio in this area may be negligible in the futurethanks to “flow-through” processes. Flow-through involves automatedsystems that can provision or maintain a circuit's path at a cost thatis nearly independent of the number of machines a circuit traverses.Accordingly, these costs may be neglected in preferred embodiments.

Finally, larger-machine portfolios consume less power than theirsmaller-machine counterparts because certain fixed costs get spread overa larger number of circuits. However, the difference in power costs issmall between portfolios and power consumption differences may beneglected from preferred embodiments.

Additional deployment cost variations responsive to element size may beapparent to one skilled in the art, and may be considered in adeployment cost analysis.

Conceptually, the deployment cost model is simple: deployment costs aresummed as they occur. A given demand profile drives the growth inequipment. The model preferably computes the number of DCS's of a chosenmatrix size required to carry the circuit demand (taking into accountcapacity expansion rules, tie pair waste, etc.). At each growth spurt, apreferred embodiment computes one or more of the following:

-   -   Either the entire cost of a new DCS or the cost of a matrix        upgrade.    -   Port costs, including both wiring (of the shelf in which line        cards reside) and equipping (line card) costs. The number of        ports added covers the additional circuit demand, including tie        pair waste and reserve port capacity.    -   A software upgrade cost.

A preferred model for analyzing deployment costs may be illustrated withthe examples shown in FIGS. 2-11. FIG. 2 shows a demand profile for a COthat grows linearly over time. The example compares four portfolios thatcarry the demand shown. Each portfolio consists of DCS's of one of fourmatrix sizes, 6000, 12000, 24000, or 36000 DS3-equivalent ports (thesecorrespond approximately to certain sizes that are presently used in theindustry).

FIG. 3 shows how many DCS's are in each portfolio in each year. In year1, the portfolios come into existence: before year 1, it can assumedthat a single 6000-port DCS served the office. In year 1, we must eitheradd another 6000-port machine or upgrade the matrix to one of the othersizes considered. Thus, in year 1 the graph shows two 6000-port DCS orone DCS of any of the other three matrix sizes. At the two portfolioextremes, it is possible to serve the maximum demand with either one DCSof the largest size or with eight DCS's of the smallest size.

The curve in FIG. 2 levels off because it shows the demand that thedifferent portfolios carry (not the offered load). Additional machineswould be necessary to carry excess demand. The DCS's in the four exampleportfolios reach capacity, thereby illustrating two different phenomenabefore and after the portfolios reach capacity. During the circuitgrowth phase, the matrices are often not full, making outage costssmaller than when the DCS reaches capacity. The deployment cost savings(of a portfolio of large machines relative to a portfolio of smallmachines) occur during this phase. After the machines fill, the annualdeployment cost savings become small (software only), but the outageimpact is greatest. The outage costs become large only after thedeployment cost savings occur. However, the discounting in present valuecalculations diminishes the effect of outage costs.

In a preferred embodiment, an expression for the cumulative, discounteddeployment cost is necessary. Denote by m a matrix size (which indexesportfolios; we shall henceforth speak of “portfolio m”), r a discountrate or cost of capital (for calculating present values), and t time.Given a portfolio with fixed matrix size m, let {s_(n): n=1, 2, . . . }be the sequence of times at which deployment costs occur, with s_(i) thetime in year 1 when a 6000-port machine is added or its matrix isupgraded. Let m_(t)=#{n:s_(n)≦t} be the number of deployment cost eventsup to time t and let q_(n) be the cost of the nth event. Then thecumulative, discounted (to time 1) total deployment cost up to time t isgiven by

$\begin{matrix}{{c_{d}( {m,r,t} )} = {\sum\limits_{n = 1}^{m_{t}}{{q_{n}( {1 + r} )}_{n}^{{- s} + 1}.}}} & (1)\end{matrix}$

Because q_(n) or s_(n) are more explicitly presented, the right handside of (1) is not encumbered with indices showing the matrix size m.Equation (1)'s purpose is to illustrate how to calculate cumulative,discounted deployment costs. In practice, time is preferably discrete(in years): then s_(n)=n (i.e., accumulate all events over a wholeyear), m_(t)=t, and q_(n) is the total cost incurred in year n.

Note that, given the demand profile, the sequence {s_(n): n=1, 2, . . .} is a deterministic process, not a stochastic one. Similarly, the costsq_(n) are not random.

FIG. 4 illustrates the deployment cost (1) in the present example as afunction of both m and t with r fixed. The x-axis displays the time tand the different curves correspond to the four values of matrix size min the example. The economies of scale manifest themselves in thedecreasing values of deployment cost as the matrix size in theportfolios increases. Symbolically, if m<m′, thenc_(d)(m,r,t)≧c_(d)(m′,r,t). For example, the four values in year 15 arethe present values of the total lifecycle cost of the four portfolios.Portfolio 36000 (with one machine) is about $5M or 20% cheaper than theportfolio 6000 (with 8 machines in year 15).

Next will be described a preferred embodiment for modeling outage costs.The model for outages differs from the deployment cost model because ofthe random nature of outages. In a preferred embodiment, the outage costhas two important attributes: the outage cost is a random variable andit is comparable to c_(d) because it is expressed in dollars.

A preferred embodiment of the general outage cost function may bedescribed as follows:

-   -   Let {S(m, n): n=1, 2, . . . } be the sequence of times at which        outages occur in portfolio m, aggregated over all DCS's in the        portfolio. The process {S(m, n)} is a stochastic process that we        expect to be nonstationary in light of demand growth (and hence        growth in the number of DCS's in the portfolio) over time.    -   Let N(m, t)=#{n:S(m, n)≦t} be the cumulative number of outages        up to time t.    -   Let x(m,t) be the number of circuits on each DCS in portfolio m        at time t. For simplicity, it may be assumed that x(m,t) is the        same for all DCS's in the portfolio.    -   Let D(m, n) be the duration of the nth outage in portfolio m.        The duration D is a random variable.    -   Let f(x, d) be the (dollar) cost of an outage of duration d that        affects x circuits. The cost does not depend on the portfolio.        The functional form off will be described later.

The cumulative, discounted (to time 1) total outage cost up to time t isdefined by

$\begin{matrix}{{C_{o}( {m,r,t} )} = {\sum\limits_{n = 1}^{N{({m,t})}}{{f( {{x( {m,{S( {m,n} )}} )},{D( {m,n} )}} )}{( {1 + r} )^{{- {S{({m,n})}}} + 1}.}}}} & (2)\end{matrix}$

The form of (2) embodies an assumption that outage cost is additive overseparate outages. This is an idealization, as a cluster of outages mighthave an impact larger than if the outages were dispersed over severalyears.

The outage cost C_(o)(m, r, t) is a random variable because all of theupper case quantities on the right hand side of (2) are randomvariables. The use of uppercase letters represents random variables andthe use of lower case letters represents deterministic quantities.

In a preferred embodiment, it is assumed that the failurecharacteristics of all DCS's are the same and do not change over time.Calculations using (2) will be in discrete time (with units a year). Indiscrete time, the number of DCS's in a portfolio will be constantthroughout the year, as will the number of circuits on each DCS. Theoutage occurrence times S(m, n) are preferably rounded to the nearestyear, which means that, in effect, the number of outages per year arecounted. In a preferred embodiment, these additional assumptions aremade:

1. Each DCS, no matter what its size, fails at most once in a year. Thefailures are governed by Bernoulli trials, which are independent betweendifferent years and between different DCS's. The probability of anoutage is a constant p. Thus, the total number of outages in a givenyear is a binomially distributed random variable, equal to the number ofheads in k biased coin tosses with head probability p, where k is thenumber of DCS's in the portfolio.

2. The outage durations D(m, n) are independent and identicallydistributed as both m and n vary.

3. The Bernoulli trials and the duration random variables areindependent.

Assumption 1 is justified because DCS's fail very infrequently; FCC dataindicates that p is roughly 0.03. If we were to assume that anindividual machine fails according to a renewal process or go one stepfurther and assume a Poisson process, the probability that the machinewill fail more than once per year is negligible. The Poisson model wouldimply that failures in different years are independent, leading to afailure model that is essentially the Bernoulli model.

Assumption 1 also says that DCS's of different sizes fail at the samerate p. This assumption is reasonable, first because the different sizeDCS's are made out of the same hardware and software, so completemachine failures with machine-related causes are independent of size.Second, all failure modes are considered, including procedural errorsand natural disasters. These also do not depend on the machine size.

According to Assumption 2, machine recovery times after outagesprobabilistically do not depend on the size of the machine. In the caseof machine-related causes, the justification here is that DCS's todayare highly modular and modules recover in parallel. Thus, the recoverytime does not depend on size.

The discrete time model facilitates computation without real loss ofgenerality. In discrete time, (2) becomes

$\begin{matrix}{{{C_{o}( {m,r,t} )} = {\sum\limits_{y = 1}^{t}{( {1 + r} )^{{- y} + 1}{\Gamma( {m,y} )}}}},} & (3)\end{matrix}$Where

$\begin{matrix}{{\Gamma( {m,y} )} = {\sum\limits_{i = 1}^{F{({m,y})}}{f( {{x( {m,y} )},{D( {m,{i + {N( {m,{y - 1}} )}}} )}} )}}} & (4)\end{matrix}$

is the outage cost in year y. In (4), the random variable F(m, y) is thebinomially distributed number of outages in portfolio m in year y.Having made the process (2) piecewise stationary (in each year) we canuse Wald's identity (Cooper, R. B., 1981, Introduction to QueueingTheory, North-Holland, New York, p. 30) on (4) to calculate the mean of(3):

$\begin{matrix}{{{EC}_{o}( {m,r,t} )} = {\sum\limits_{y = 1}^{t}{{{EF}( {m,y} )}{{Ef}( {{x( {m,y} )},D} )}{( {1 + r} )^{{- y} + 1}.}}}} & (5)\end{matrix}$

Here the duration random variable has been abbreviated to the generic D,by invoking Assumption 2. E denotes the mathematical expectation.

The outage cost measures risk with the same “yardstick” (dollars) thatwe use to measure deployment cost. A functional form for the cost f ofan outage may be determined by decomposing f into two summands, directcosts such as lost revenue f_(LR) and indirect costs such as “rippleeffects” cost f_(RE). That is, f_(LR)=f_(LR)+f_(RE).

In the example described above, lost revenue has two components,corresponding to voice and data traffic. When a DCS fails, the carrierdoes not collect toll revenue for the interoffice calls that cannotcomplete because of the outage. On top of that, FCC tariffs specify thatcustomers with special access circuits (data “pipes” like DS1's andDS3's) can claim “credit allowances” if they lose service for more thana specified amount of time. For example, in FCC Tariff 11 the creditallowance is a full month's charges after one minute of outage time.

In addition, there are certain repercussions of an outage, especially ifthe outage is a serious one. When an outage occurs, wireline voicecallers in the midst of calls are cut off and new callers cannotcomplete calls. These customers may dial again (they may find analternate route through a working DCS) or try cell phones. Such actionsmay create overloads on other network elements and lead to furtherservice disruption.

A significant outage may also damage the carrier's reputation, leadingto long-term revenue loss. The carrier could face increased regulatoryscrutiny. All of these repercussions may be termed “ripple effects.”While they may be difficult to quantify, they may create a significantportion of the risk that large network elements present. We now discusslost revenue and ripple effects in turn.

We will sketch the form of the summand f_(LR) off corresponding to lostrevenue and the assumptions used to arrive at the lost revenue functionin the current example.

Broadband DCS's carry two kinds of circuits, voice trunks and specialaccess circuits. Assume that the percentage of circuits of each kind onthe DCS is known.

When a DCS fails, some or all of the trunks between central offices andtandems no longer work. If the failure of the trunks leads to lostcalls, the carrier gets no revenue from those calls. The lost voicerevenue may be computed as the product of three factors, the averagerate at which calls are lost, the revenue per call, and the outageduration. The last of these is the second argument of f. The revenue percall may be readily estimated from typical regional long distance ratesand typical holding times.

The call loss rate is somewhat more complex. As an example, suppose thatthe trunk group from a CO to a tandem is split equally between two DCS'sand one DCS fails. From the trunk group's utilization before the outage,we can infer the call attempt rate and the percentage of calls lostafter the outage (e.g., by using the Erlang B formula (Cooper, 1981)).The product of the attempt rate and the percentage of calls lost givesthe call loss rate for the CO. We then sum over affected CO's, which wecan do indirectly by counting DS3's that carry voice trunks.

Taking an average value for the pre-failure trunk utilization andaverage revenue per call figures makes the revenue lost from voice callslinear in D and x, the total number of circuits through the DCS.

Tariff FCC 11 states that if the outage duration D exceeds a threshold d(one minute), then affected special access circuit customers may demanda credit allowance. To calculate the actual allowance paid out, firstsum over circuit types the number of circuits times the monthly chargefor the circuit. Multiply this by the percentage of customers who demanda credit allowance and by the indicator I(D>d), which takes the value 1if D>d and 0 otherwise. Given fixed percentages of special circuits ofeach type, the resulting lost revenue from special circuits is linear inx and I(D>d).

Putting the results for voice and specials together, an expression forlost revenue isf _(LR)(x,D)=x(vD+sI(D>d))  (6)

Here, v and s are voice and special access circuit (constant)coefficients.

Given the subjective nature of ripple effects, a suitable way toestimate their cost may be through a series of structured interviewswith executives who make significant decisions as a result of majoroutages. The interviewers may ask the executives to compare theseverities of various outages that differ in numbers of circuitsaffected and in duration and to express the answers in dollar terms.Additional ways of assessing ripple effect cost are apparent to thoseskilled in the art. The result is an estimate for the ripple effect costf_(RE)(x, d) as a function of circuits x and duration d.

A key feature of “ripple effects” is that if one outage is twice aslarge as another, then the impact of the larger outage is more thantwice as large as the impact of the smaller outage. We can capture thisbehavior by assuming that the function g(x)=Ef(x, D) is convex. By anelementary fact about convex functions (Royden, H. L., 1968. RealAnalysis, Macmillan, New York, p. 108) and the fact that g(0)=0, g(x)/xis increasing in x, doubling the size x of an outage more than doublesits cost. The convexity of g also implies that (5), which equals

$\begin{matrix}{{{EC}_{o}( {m,r,t} )} = {\sum\limits_{y = 1}^{t}{{{EF}( {m,y} )}{g( {x( {m,y} )} )}( {1 + r} )^{{- y} + 1}}}} & (7)\end{matrix}$

increases with m. To see this, let n(m, y) be the number of DCS's inportfolio m in year y; F(m, y) is the number of successes in n(m, y)Bernoulli trials with success probability p. Thus, EF(m, y)=n(m, y)p.Write (7) as

${{EC}_{o}( {m,r,t} )} = {p{\sum\limits_{y = 1}^{t}{{n( {m,y} )}{{x( {m,y} )}\lbrack {{g( {x( {m,y} )} )}/{x( {m,y} )}} \rbrack}{( {1 + r} )^{{- y} + 1}.}}}}$

Consider the product k(m, y)=n(m, y)x(m, y) of the number of DCS inportfolio m and the number of circuits on each DCS in year y. It istempting, but not quite true, to say that k(m, y) is the total number ofcircuits served in year y and therefore independent of m. The assertionis false because any circuit that traverses a tie pair gets countedtwice, making k(m, y) slightly larger than the total number of circuitswhen the portfolio has more than one DCS. As a result, k(m, y) isslightly decreasing in m. But pretend, heuristically, that k(m, y) isindependent of m.

Next, note that x(m, y) is increasing in m. Since g(x)/x is increasingin x, the term in brackets increases with m and it follows that thatEC_(o)(m,r,t) is increasing in m.

To make g convex, assume that f_(RE) is separable in the sense thatthere are functions f_(RE,S) and f_(RE,D) such thatf_(RE)(x,d)=f_(RE,S)(x)f_(RE,D)(d). Combining this with (6) yieldsg(x)=x(vED+sP(D>d))+f _(RE,S)(x)Ef _(RE,D)(D).

Note that duration affects average ripple costs only as a constantmultiplier because outage durations have the same distribution for allportfolios. If we assume that f_(RE,S) is convex, then clearly g isconvex.

In the discussion that follows, take f_(RE,D)(z)=vz+sI(z>d), the samecost coefficient as in the lost revenue, and choose f_(RE,S) to be afunction that is zero for x<2000 and increases quadratically abovex=2000. The intent is to have the ripple effect cost increasedramatically compared to lost revenue for large outages. FIG. 5 comparesthe lost revenue to two choices for f_(RE,S). Ripple effect B is chosento be larger than ripple effect A. For reference, the figure alsoincludes lost revenue.

FIG. 6 plots EC_(o)(m,r,t) over time for the four portfolios in ourrunning example, using ripple effect A costs. It shows that, for anygiven time horizon t, EC_(o)(m,r,t) increases with m.

Total cost refers to the sum of deployment cost and outage cost (bothcumulative and discounted): C(m,r,t)=c_(d)(m,r,t)+C_(o)(m,r,t). Thetotal cost is a random variable because the outage cost is random.

Average total cost is c(m,r,t)=EC(m,r,t)=c_(d)(m,r,t)+EC_(o)(m,r,t). Asshown in FIG. 4, c_(d)(m,r,t) is decreasing in m and, based on (7),EC_(o)(m,r,t) is increasing in m. The same arguments as for (7), withthe expectation of (6) in place of g, would show that EC_(o)(m,r,t) isroughly constant with respect to m when the outage cost is linear in x.In this case, c(m,r,t) would decrease with m. That is, if outage costsare linear in x, a bigger DCS is better, as measured by average totalcost. With the nonlinear costs resulting from ripple effects, thebehavior of c(m,r,t) as m grows is not clear.

FIG. 7 shows graphs of c(m,r,t) with respect to m when t is set to 15years (the life cycle). The bottom curve corresponds to an outage costconsisting of lost revenue alone (zero ripple effect costs). As justdiscussed, the average total cost decreases as m increases, reflectingthe economies of scale of large machines and the fact that average lostrevenue is the same for all portfolios.

The other two curves in FIG. 7 correspond to ripple effects A and B.Average total cost B increases over the entire range of matrix sizes,while average total cost A is relatively flat, increasing a bit fromportfolio 24000 to portfolio 36000. The increase comes from the stiffcost penalty that the quadratic ripple effect functions (B especially)in FIG. 5 attach to large outages.

The average total cost suggests that there is a size that is “too big.”In the case of ripple effect A, the 36000-port matrix is too big becauseportfolio 36000's average total cost exceeds that portfolio 24000. Underripple effect B, every matrix shown is too big and some matrix smallerthan 6000 ports is optimal with respect to average total cost. However,before drawing any conclusions based on averages, it is preferable toconsider the variability of total cost.

Fluctuations in the annual outage cost F(m, y) may occur for tworeasons. First, outages may or may not occur. Second, when outagesoccur, their durations may vary. Randomness in occurrences is generallymore important than randomness of duration. As m increases, wedistribute the same total number of circuits over fewer DCS's. Ineffect, we make fewer but bigger “bets” (that the DCS will not fail)with the same total amount of “cash” (circuits). A portfolio of bigDCS's will experience fewer failures per year (it has fewer DCS's tofail), but each failure will affect more circuits. On the other hand,the outage duration affects all portfolios equally.

The coefficient of variation of F(m, y) conveniently formalizes thisargument, in light of the assumed separability of our cost functions. Asdescribed above, f(x, z)=f_(s)(x)(vz+sI(z>d)), wheref_(s)(x)=x+f_(RE,S)(x). Since the threshold d in FCC Tariff 11 is onlyone minute and all the outages in our FCC data exceed 30 minutes, wewill replace I(z>d) by 1, yielding f(x, z)=f_(s)(x)(vz+s). Substitutingthis into (4) and using EF(m, y)=n(m, y) p, gives a mean annual outagecost ofEΓ(m,y)=n(m,y)pf _(s)(x(m,y))(vED+s).

For the variance of Γ(m, y) we use Wald's identity for second moments(Cooper, 1981, p. 30) and the result Var F(m,y)=n(m, y)p(1−p) to getVarΓ(m,y)=n(m,y)pf _(s)(x(m,y))² [v ²VarD+(1−p)(vED+s)²]

Then the coefficient of variation (standard deviation divided by themean) of Γ(m, y) is

${{CV}\;{\Gamma( {m,y} )}} = {\frac{1}{ \sqrt{}( {{n( {m,y} )}p} ) }\frac{ \sqrt{}( {{v^{2}{Var}\; D} + {( {1 - p} )( {{vED} + s} )^{2}}} ) }{{vED} + s}}$

Given our data, the second fraction here (involving the moments of D) isabout 1.15. However, since p=0.03, the first factor 1/√(n(m, y)p) rangesfrom 2 to 6 as n(m, y) ranges from 8 to 1. Thus, the rarity of outagesis the dominant contributor to the coefficient of variation of Γ(m, y)and it makes the fluctuations in Γ(m, y) larger than the mean by afactor between 2 and 6.

The outage cost C_(o)(m,r,t) in (3) inherits the fluctuations in Γ(m,y). Although the summation and discounting in (3) smoothes out some ofthe fluctuations, the coefficient of variation of C_(o)(m,r,t) can stillbe large, even for those portfolios with the smallest m.

Our assumptions about the independence of outages in different yearsimply that

${{Var}\;{C_{o}( {m,r,t} )}} = {\sum\limits_{y = 1}^{t}{( {1 + r} )^{{{- 2}y} + 2}{Var}\;{\Gamma( {m,y} )}}}$

FIG. 8 compares the cumulative, discounted outage cost standarddeviations using ripple effect costs A (compare to FIG. 6). Portfolio6000 has a mean lifecycle outage cost of $2.4M and a standard deviationof $2.0M, yielding a coefficient of variation of 0.8. At the otherextreme, portfolio 36000 has a mean of $5.9M and a standard deviation of$10.8M, yielding a coefficient of variation of 1.8.

Note that these results apply to portfolios serving one CO only. Inreality, a large carrier puts broadband DOS's in many CO's. Let L be thenumber of locations in which the carrier will adopt a policy ofdeploying broadband DOS's of one of the four sizes in our runningexample. Assuming for the sake of illustration that the demand is thesame in all L locations, the outage cost becomes the sum of Lindependent copies of C_(o)(m,r,t). Since the mean and variance of theoutage cost are both L times that of C_(o)(m,r,t), the coefficient ofvariation of the resulting outage cost equals the coefficient ofvariation of C_(o)(m,r,t) divided by A.

Although averaging over a larger number of central offices will decreasethe outage cost fluctuations relative to their means, we continue tolook at a single office, realizing that risk results corresponding to asingle office are higher than those for more offices.

The fluctuations of the total cost are the same as those of the outagecost because the total cost consists of the outage cost plus thenon-random deployment cost. If ripple effects resemble A more than theydo B, the average total cost EC(m,r,t) does not change much as m varies(FIG. 7); fluctuations in outage cost could easily make a portfolio thatis cheaper on average considerably more expensive in an actual outcome.For example, using ripple effect A, the total expected cost of theportfolio 6000 is $30.1M and the standard deviation is $2.0M. Bycomparison, the total expected cost of portfolio 36000 is $29.3M, butits standard deviation is $10.8M. A typical fluctuation above the meancould easily push the cost of the latter well above that of the former.

FIG. 9 presents histograms of the lifecycle cost distributions of allfour portfolios. Three features are apparent. First, the larger thevalue of m, the longer the tail of the distribution. Second, largervalues of m correspond to smaller minimum costs. For example, as thematrix increases from 6000 ports to 36000 ports, the corresponding fourminimum lifecycle costs are $27.6M, $25.6M, $23.8M, and $22.7M. Theminimum values are the lifecycle deployment costs (i.e., total costoutcomes where the outage cost is zero). Third, the larger m is, themore likely is an outcome with minimum cost. The portfolio with largerDCS's has fewer DCS's and therefore fewer Bernoulli trials to generateoutages. With fewer trials, the probability of zero outages in alifecycle is greater. The probabilities of no outages over a lifecycleare, in order of increasing matrix size, 11%, 29%, 51%, and 63%.

These features make it hard to decide intuitively which portfolio is“best.” A portfolio of larger DCS's offers a reward that is in the formof a lower minimum cost and a higher probability of attaining theminimum (and sometimes a lower mean). But the risk is that a cost farout in the tail will occur: concentration of circuits onto a smallnumber of DCS's makes catastrophes possible.

To settle the risk-reward tradeoff, the notion of stochastic dominancemay be invoked. The literature on economic applications of stochasticdominance typically concerns random variables where larger values arebetter, as opposed to costs, where larger is worse. To switch to thestandard point of view, it is preferable to look at profits. Because allportfolios serve exactly the same traffic demands, the discountedrevenues that the portfolios bring in over a lifecycle have a common(non random) value r. The revenue minus the total cost r−C(m,r,t) is theappropriate random variable.

To define stochastic dominance, denote by X and Y two different totallifecycle costs (i.e., C(m,r,t) for two different values of m). Adaptingthe definitions of Ogryczak, W. and Ruszczyński, A., 1999. Fromstochastic dominance to mean-risk models: semideviations as riskmeasures, European Journal of Operational Research, 116 (1), 33-50, wesay that r−Y exhibits first-degree stochastic dominance (FSD) over r−Xand writer−Y≧ _(FSD) r−X if and only if P(X>ξ)≧P(Y>ξ) for all real ξ

We say that r−Y exhibits second-degree stochastic dominance (SSD) overr−X and writer−Y≧ _(SSD) r−X if and only if E(X−ξ)⁺ ≧E(Y−ξ)⁺ for all real ξ

Here (x)⁺=max(x,0) denotes the positive part of x. Note that by settingξ=0, we get E(X)≧E(Y) since costs are nonnegative.

FSD implies SSD, as can be seen by substituting a random variable for xand applying the expectation operator and Fubini's theorem to theidentity

(x − ξ)⁺ = ∫_(ξ)^(∞)I(x > z) 𝕕z.

As a result of the first two features mentioned in connection with FIG.9, our total lifecycle costs are not ordered by FSD. For if we take X tocorrespond to a larger value of m than Y, the first feature says thatP(X>ξ)≧P(Y>ξ) for all sufficiently large ξ while the second says thatmin X<min Y. But P(X>ξ)<P(Y>ξ)=1 for min X<ξ<min Y.

Calculations with the distribution functions of the four different totallifecycle costs show that the costs are ordered according to SSD.Portfolio 12000 has the lowest cost and is slightly lower than portfolio24000. Portfolios 6000 and 36000 have the highest costs.

Stochastic dominance has well-known connection with utility functions. Autility function (Luce, R. D., Raiffa, H., 1957. Games and Decisions,John Wiley & Sons, Inc., New York) is a deterministic function u suchthat u(z) is the subjective value of a reward z, such as an investmentreturn. The expected utility of a random payoff resolves the risk-rewardtradeoff. For example, Eu(r−C(m,r,t)) weighs the reward of a loweraverage or minimum cost against the risk of large deviation above theaverage. Because u encodes the decision maker's perceptions of thesignificance of lifecycle costs, portfolio m₁ is better than portfoliom₂ if and only if its expected utility is higher:Eu(r−C(m₁,r,t))>Eu(r−C(m₂,r,t)).

Utility functions in economics are frequently increasing and concave.They increase because a bigger reward is better and are concave becauseof the decreasing marginal utility of wealth. Assumingdifferentiability, the two properties of the utility function u are u′≧0and u″≦0. The connection between SSD and utility functions is that ifr−Y≧_(SSD)r−X, then for any u with u′≧0 and u″≦0, Eu(r−Y)>Eu(r−X) (Levy,H., 1992. Stochastic dominance and expected utility: survey andanalysis, Management Science, Vol. 38, No. 4, 555-593; Ogryczak, 1999).

We now illustrate the application, of utility to portfolio selectionwith linear transformations of utility functions of the basic formu(z)=z−α exp(−βz) (Bell, D. E., 1995. Risk, return, and utility,Management Science, Vol. 41, No. 1, 23-30), where a and fl are positiveconstants. We will consider the dependence of utility on cost. Revenuedoes not play an explicit role.

FIG. 10 shows one example utility function (of cost). We normalize theutility to zero at the expected total cost ($30.1 M) of portfolio 6000.Costs less than $30.1M carry positive utility, while costs larger havenegative utility. The utility decreases exponentially to −0.0 as thecost grows. This behavior penalizes a cost distribution with a longtail.

FIG. 11 shows the expected utilities of the four portfolios, using theutility function in FIG. 10 and total costs resulting from ripple effectA. Portfolio 6000 and portfolio 36000 have lower utility than the middletwo portfolios. Portfolio 6000 has lower utility because it has thehighest deployment costs. The long tails of portfolio 36000's costdistribution give it the lowest utility. Portfolio 24000 and portfolio12000 have similar means, but portfolio 24000 has a somewhat longertail, giving it a slightly lower utility. The utility values are orderedaccording to SSD.

Presented with the evidence in FIG. 11, the decision maker wouldconclude that the rewards of bigger DCS's (economies of scale) outweighthe outage risks if the carrier is contemplating an increase in matrixsize from 6000 to 12000 ports. A further increase in matrix size to24000 ports has little effect on the risk-reward tradeoff. However,portfolio 36000 is “too big,” for the risk of a catastrophic outage nowoutweighs the rewards of greater size.

Note that average total cost in FIG. 7 provides the same qualitativeconclusions (as noted above, the means follow the SSD ordering) but theutility function is less equivocal about portfolio 36000.

Another preferred embodiment of the present invention is shown in FIG.12. The first step in the embodiment shown in FIG. 12, step 1201, thedeployment cost for each of a plurality of portfolios is determined. Inthe second step, step 1202, the average outage cost is determined. Inthe third step, step 1203, the average total cost is determined bysumming deployment cost and average outage cost. Next, in step 1204, thetotal cost variability is determined. A procedure for determining thevariability of total cost is described above. Next, in step 1205, theexpected utility for at least two portfolios is determined. As describedabove, expected utility is determined responsive to total costvariability and an assumed utility function. Finally, in step 1206, thedetermined expected utilities of the portfolios are compared. Ingeneral, it would be advisable for a decision maker to select theportfolio with the highest expected utility.

Another preferred embodiment of the present invention is shown in FIG.13. Similar to the embodiment shown in FIG. 12, in the first step in theembodiment shown in FIG. 13, step 1301, the deployment cost for each ofa plurality of portfolios is determined. In the second step, step 1302,the average outage cost is determined. In the third step, step 1303, theaverage total cost is determined by summing deployment cost and averageoutage cost. Next, in step 1304, the total cost variability isdetermined. A procedure for determining the variability of total cost isdescribed above. If the total cost variability is greater than somepredetermined threshold value, then the preferred embodiment proceeds tostep 1305, wherein the expected utility for at least two portfolios isdetermined. As described above, expected utility is determinedresponsive to total cost variability and an assumed utility function.Then, in step 1306, the determined expected utilities of the portfoliosare compared. In general, it would be advisable for a decision maker toselect the portfolio with the highest expected utility. If the totalcost variability is not greater than the particular predeterminedthreshold value, the expected utility is not determined. Rather, thepreferred embodiment proceeds to step 1307, wherein the average totalcost of at least two of the portfolios is compared. In thiscircumstance, it would generally be advisable for a decision maker toselect the portfolio with the lowest total cost. The predeterminedthreshold value for the total cost variability may be any appropriatevalue selected by one skilled in the art. In one embodiment, thepredetermined threshold value may be 0.

In another preferred embodiment, the invention is carried out in acomputer environment. Thus, for example, a computer program product maybe provided comprising a computer usable medium having computer logicrecorded thereon for instructing a computer to carry out the stepsdescribed, for example, with respect to either FIG. 1, 12 or 13.

FIG. 14 is a block diagram of a computer system as may be used toimplement an embodiment of the invention.

With reference now to FIG. 14, a description of a computer systemsuitable for use with an embodiment of the present invention isprovided. The computer system 1402 includes one or more processors, suchas a processor 1404. The processor 1404 is connected to a communicationbus 1406.

The computer system 1402 also includes a main memory 1408, preferablyrandom access memory (RAM), and can also include a secondary memory1410. The secondary memory 1410 can include, for example, a hard diskdrive 1412 and/or a removable storage drive 1414, representing a floppydisk drive, a magnetic tape drive, an optical disk drive, etc. Theremovable storage drive 1414 reads from and/or writes to a removablestorage unit 1418 in a well-known manner. The removable storage unit1418, represents a floppy disk, magnetic tape, optical disk, etc. whichis read by and written to by the removable storage drive 1414. As willbe appreciated, the removable storage unit 1418 includes a computerusable storage medium having stored therein computer software and/ordata.

In alternative embodiments, the secondary memory 1410 may include othersimilar means for allowing computer programs or other instructions to beloaded into the computer system 1402. Such means can include, forexample, a removable storage unit 1422 and an interface 1420. Examplesof such can include a program cartridge and cartridge interface (such asthat found in video game devices), a removable memory chip (such as anEPROM, or PROM) and associated socket, and other removable storage units1422 and interfaces 1420 which allow software and data to be transferredfrom the removable storage unit 1422 to the computer system 1402.

The computer system 1402 can also include a communications interface1424. The communications interface 1424 allows software and data to betransferred between the computer system 1402 and external devices.Examples of the communications interface 1424 can include a modem, anetwork interface (such as an Ethernet card), a communications port, aPCMCIA slot and card, etc. Software and data transferred via thecommunications interface 1424 are in the form of signals 1426 that canbe electronic, electromagnetic, optical or other signals capable ofbeing received by the communications interface 1424. Signals 1426 areprovided to communications interface via a channel 1428. A channel 1428carries signals 1426 and can be implemented using wire or cable, fiberoptics, a phone line, a cellular phone link, an RF link and othercommunications channels.

In this document, the term “computer-readable storage medium” is used togenerally refer to media such as the removable storage device 1418, ahard disk installed in hard disk drive 1412, and signals 1426. Thesemedia are means for providing software and operating instructions to thecomputer system 1402.

Computer programs (also called computer control logic) are stored in themain memory 1408 and/or the secondary memory 1410. Computer programs canalso be received via the communications interface 1424. Such computerprograms, when executed, enable the computer system 1402 to perform thefeatures of the present invention as discussed herein. In particular,the computer programs, when executed, enable the processor 1404 toperform the features of the present invention. Accordingly, suchcomputer programs represent controllers of the computer system 1402.

In an embodiment where the invention is implemented using software, thesoftware may be stored in a computer-readable storage medium and loadedinto the computer system 1402 using the removable storage drive 1414,the hard drive 1412 or the communications interface 1424. The controllogic (software), when executed by the processor 1404, causes theprocessor 1404 to perform the functions of the invention as describedherein.

In another embodiment, the invention is implemented primarily inhardware using, for example, hardware components such as applicationspecific integrated circuits (ASICs). Implementation of such a hardwarestate machine so as to perform the functions described herein will beapparent to persons skilled in the relevant art(s). In yet anotherembodiment, the invention is implemented using a combination of bothhardware and software.

Although particular embodiments of the invention have been described andillustrated herein, it is recognized that modifications and variationsmay readily occur to those skilled in the art and consequently it isintended that the claims be interpreted to cover such modifications andequivalents.

The invention claimed is:
 1. A processor-implemented method foranalyzing risk comprising: determining, via a processor of a computingdevice, a deployment cost for each of a plurality of communicationnetwork element portfolios, wherein each communication network elementportfolio comprises one or more communication network elements with apredetermined communication capacity; determining, via the processor, anoutage cost for each of the plurality of communication network elementportfolios responsive to the probability of one or more communicationnetwork element outages over a predetermined life cycle period;determining, via the processor, a total cost for each of the pluralityof communication network element portfolios, wherein total cost isdetermined by summing the deployment cost and the outage cost for eachof the plurality of communication network element portfolios, whereinthe deployment cost and the outage cost are expressed in comparableunits of measure; and comparing, via the processor, the total cost of afirst communication network element portfolio of the plurality ofcommunication network element portfolios with the total cost of a secondcommunication network element portfolio of the plurality ofcommunication network element portfolios, wherein the total cost is anindicator of the deployment cost and the risk.
 2. The method of claim 1,wherein each communication network element of the first communicationnetwork element portfolio has a first predetermined communicationcapacity, and each communication network element of the secondcommunication network element portfolio has a second predeterminedcommunication capacity.
 3. The method of claim 2, wherein the firstpredetermined communication capacity is different from the secondpredetermined communication capacity.
 4. The method of claim 1, whereineach communication network element of the first communication networkelement portfolio has a substantially identical predeterminedcommunication capacity.
 5. The method of claim 4, wherein the number ofcommunication network elements in the first communication networkelement portfolio is determined responsive to the substantiallyidentical predetermined communication capacity and a total communicationcapacity of the first communication network element portfolio.
 6. Themethod of claim 1, wherein the deployment cost for the firstcommunication network element portfolio is determined responsive to thecost of obtaining each communication network element in the firstcommunication network element portfolio and a recurring cost associatedwith maintaining each communication network element in the firstcommunication network element portfolio over a predetermined timeperiod.
 7. The method of claim 1, wherein the outage cost for the firstcommunication network element portfolio comprises a direct cost and anindirect cost associated with the one or more communication networkelement outages of the first communication network element portfolio. 8.The method of claim 7, wherein the direct cost comprises lost revenueassociated with the one or more communication network element outages.9. The method of claim 7, wherein the direct cost is determinedresponsive to the predetermined communication capacity of the one ormore communication network elements in the first communication networkelement portfolio.
 10. The method of claim 7, wherein the indirect costis a predetermined function responsive to the predeterminedcommunication capacity of the one or more communication network elementsin the first communication network element portfolio.
 11. The method ofclaim 10, wherein the predetermined function has the characteristic thata doubling of the predetermined communication capacity of the one ormore communication network elements in the first communication networkelement portfolio more than doubles the indirect cost associated with atleast one of the communication network element outages of the firstcommunication network element portfolio.
 12. The method of claim 1,wherein each of the plurality of communication network elementportfolios comprises one or more telecommunications network elements.13. The method of claim 12, further comprising: repeating the step ofdetermining the deployment cost if an additional telecommunicationsnetwork element is added to a communication network portfolio, or thecommunication capacity of the one or more telecommunications networkelements that comprise the communication network element portfolio ischanged.
 14. The method of claim 12, wherein the step of determining thedeployment cost is carried out responsive to one or more of the groupconsisting of, a cost of obtaining each telecommunications networkelement, a cost of upgrading each telecommunications network element, acost of updating software, and a port cost.
 15. The method of claim 12,wherein the outage cost for at least one of the plurality ofcommunication network element portfolios comprises lost revenue andripple effect costs.
 16. The method of claim 1, wherein each of theplurality of communication network element portfolios comprises adigital cross-connect.
 17. A processor-implemented method for analyzingrisk comprising: determining, via a processor of a computing device, adeployment cost for each of a plurality of communication network elementportfolios, wherein each communication network element portfoliocomprises one or more communication network elements with apredetermined communication capacity; determining, via the processor, anaverage outage cost for each of the plurality of communication networkelement portfolios responsive to the probability of communicationnetwork element failure; determining, via the processor, an averagetotal cost for each of the plurality of communication network elementportfolios, wherein average total cost is determined by summing thedeployment cost and the average outage cost for each of the plurality ofcommunication network element portfolios and wherein the deployment costand the average outage cost are expressed in comparable units ofmeasure; determining, via the processor, a total cost variability foreach of the plurality of communication network element portfolios;determining, via the processor, an expected utility for a first and asecond of the plurality of communication network element portfolios,wherein the expected utility is determined responsive to the total costvariability and a utility function; and comparing, via the processor,the expected utility of the first and second communication networkelement portfolios, wherein the expected utility is an indicator of thedeployment cost and the risk.
 18. The method of claim 17, wherein eachof the plurality of communication network element portfolios comprisesone or more telecommunications network elements.
 19. The method of claim18, further comprising: repeating the step of determining the deploymentcost if an additional telecommunications network element is added to acommunication network element portfolio, or the communication capacityof the one or more telecommunications network elements that comprise thecommunication network element portfolio is changed.
 20. The method ofclaim 18, wherein the step of determining the deployment cost is carriedout responsive to one or more of the group consisting of, a cost ofobtaining each telecommunications network element, a cost of upgradingeach telecommunications network element, a cost of updating software,and a port cost.
 21. The method of claim 18, wherein the outage cost forat least one of the plurality of communication network elementportfolios comprises lost revenue and ripple effect costs.
 22. Themethod of claim 17, wherein each of the plurality of communicationnetwork element portfolios comprises a digital cross-connect.
 23. Themethod of claim 17, wherein each of the plurality of communicationnetwork element portfolios has a substantially identical revenue. 24.The method of claim 23, wherein the utility function is responsive to aperceived value of the average total cost subtracted from the revenue.25. The method of claim 23, wherein the utility function has thecharacteristic that a first derivative of the utility function isgreater than or equal to zero and a second derivative of the utilityfunction is less than or equal to zero.
 26. The method of claim 23,wherein the utility function has the characteristic that it increases asthe perceived value of the average total cost subtracted from therevenue increases and is concave.
 27. A processor-implemented method foranalyzing communication network elements comprising: determining, via aprocessor of a computing device, a deployment cost for each of aplurality of communication network element portfolios, wherein eachcommunication network element portfolio comprises one or morecommunication network elements with a predetermined communicationcapacity; determining, via the processor, an average outage cost foreach of the plurality of communication network element portfoliosresponsive to the probability of communication network element failure;determining, via the processor, an average total cost for each of theplurality of communication network element portfolios, wherein averagetotal cost is determined by summing the deployment cost and the averageoutage cost for each of the plurality of communication network elementportfolios and wherein the deployment cost and the average outage costare expressed in comparable units of measure; determining, via theprocessor, a total cost variability for each of the plurality ofcommunication network element portfolios; determining, via theprocessor, an expected utility for a first and a second of the pluralityof communication network element portfolios, wherein the expectedutility is determined responsive to the total cost variability and autility function; and selecting, via the processor, one of the first andthe second of the plurality of communication network element portfolioswith a highest expected utility.
 28. A system for analyzingcommunication network elements comprising: means for determining adeployment cost for each of a plurality of communication network elementportfolios, wherein each communication network element portfoliocomprises one or more communication network elements with apredetermined communication capacity; means for determining an averageoutage cost for each of the plurality of communication network elementportfolios responsive to the probability of communication networkelement failure; means for determining an average total cost for each ofthe plurality of communication network element portfolios, whereinaverage total cost is determined by summing the deployment cost and theaverage outage cost for each of the plurality of communication networkelement portfolios and wherein the deployment cost and the averageoutage cost are expressed in comparable units of measure; means fordetermining a total cost variability for each of the plurality ofcommunication network element portfolios; means for determining anexpected utility for a first and a second of the plurality ofcommunication network element portfolios, wherein the expected utilityis determined responsive to the total cost variability and a utilityfunction; and means for selecting one of the first and the second of theplurality of communication network element portfolios with a highestexpected utility.
 29. The system of claim 28, wherein each of theplurality of communication network element portfolios comprises one ormore telecommunications network elements.
 30. The system of claim 29,wherein the means for determining the deployment cost re-determinesdeployment cost if an additional telecommunications network element isadded to a communication network element portfolio, or the communicationcapacity of the one or more telecommunications network elements thatcomprise the communication network element portfolio is changed.
 31. Thesystem of claim 29, wherein the means for determining the deploymentcost is responsive to one or more of the group consisting of, a cost ofobtaining each telecommunications network element, a cost of upgradingeach telecommunications network element, a cost of updating software,and a port cost.
 32. The system of claim 29, wherein the outage cost forat least one of the plurality of communication network elementportfolios comprises lost revenue and ripple effect costs.
 33. Thesystem of claim 28, wherein each of the plurality of communicationnetwork element portfolios comprises a digital cross-connect.
 34. Acomputer program product comprising a non-transitory computer usablemedium having computer program logic recorded thereon to be executed bya processor of a computing system to instruct the computer system to:determine a deployment cost for each of a plurality of communicationnetwork element portfolios, wherein each communication network elementportfolio comprises one or more communication network elements with apredetermined communication capacity; determine an average outage costfor each of the plurality of communication network element portfoliosresponsive to the probability of communication network element failure;determine an average total cost for each of the plurality ofcommunication network element portfolios, wherein average total cost isdetermined by summing the deployment cost and the average outage costfor each of the plurality of communication network element portfoliosand wherein the deployment cost and the average outage cost areexpressed in comparable units of measure; determine a total costvariability for each of the plurality of communication network elementportfolios; determine an expected utility for a first and a second ofthe plurality of communication network element portfolios, wherein theexpected utility is determined responsive to the total cost variabilityand a utility function; and compare the expected utility of the firstand second communication network element portfolios, wherein theexpected utility is an indicator of the deployment cost and the risk.35. A processor-implemented method for analyzing risk comprising:determining, via a processor of a computing device, a deployment costfor each of a plurality of communication network element portfolios,wherein each communication network element portfolio comprises one ormore communication network elements with a predetermined communicationcapacity; determining, via the processor, an average outage cost foreach of the plurality of communication network element portfoliosresponsive to the probability of communication network element failure;determining, via the processor, an average total cost for each of theplurality of communication network element portfolios, wherein averagetotal cost is determined by summing the deployment cost and the averageoutage cost for each of the plurality of communication network elementportfolios and wherein the deployment cost and the average outage costare expressed in comparable units of measure; determining, via theprocessor, a total cost variability for each of the plurality ofcommunication network element portfolios; if the total cost variabilityis greater than a predetermined value, determining, via the processor,an expected utility for a first and a second of the plurality ofcommunication network element portfolios, wherein the expected utilityis determined responsive to the total cost variability and a utilityfunction, and comparing the expected utility of the first and secondcommunication network element portfolios, wherein the expected utilityis an indicator of the deployment cost and the risk; and if the totalcost variability is not greater than the predetermined value, comparing,via the processor, the average total cost for the first and secondcommunication network element portfolios, wherein the average total costis an indicator of the deployment cost and the risk.
 36. The method ofclaim 35, wherein the predetermined value is 0.